Relational Logic (RL) and Axiomatic Context (AC) : Extending the Boundaries of Boolean and Fuzzy Paradigms

This paper introduces two novel mathematical frameworks: Relational Logic (RL) and Axiomatic Context (AC). By expanding the traditional truth value spaces from discrete sets {0, 1} or closed intervals [0, 1] to the extended real number line R¯ = R ∪ {−∞, +∞}, we redefine logic based on order comparison (“than”) rather than strict equality (“is”). We demonstrate that both classical Boolean logic and Zadeh’s Fuzzy Logic are special bounded cases within RL and AC. Furthermore, the proposed system exhibits paraconsistency, naturally absorbing contradictions as measurable dissonance signals without systemic collapse. This version introduces crucial theoretical refinements and formalizes the underlying algebraic structures of the framework to ensure strict alignment with core principles of pure mathematical analysis and measure theory. Key Updates in This Version: Formalization of AC Context-Field: Replaced all physical metaphors surrounding 'Field Energy' with the mathematically rigorous concept of AC Context-Field Total Magnitude (\(\mathcal{M}\)), establishing the framework entirely within continuous scalar field theory and unbounded measure spaces. Domain Boundary Specification: Explicitly defined the restricted sub-domains for classical logic unification, demonstrating that Boolean and Fuzzy architectures map strictly to the closed intervals \(\{-1, +1\}\) and \([-1, 1]\) respectively, while Relational Logic (RL) smoothly transcends these boundaries to accommodate transfinite states. Algebraic Decoupling and Proof Stability: Formalized the operational layer decoupling within Theorem 3.2. This explicitly clarifies that the non-associativity of the signed composition operator governs only causal contextual state transitions, while the final integrated field spatial totalization remains completely path-invariant and globally convergent.